Spacetime, Phil. Spacetime

Mach's Principle
Ernst Mach (1838-1916)

What is "Mach's Principle"? This word is coined by Einstein in 1918. As you will see, it is hard to give an unequivocal answer. This principle is said to have played an important role in forming Einstein's general relativity, although the final theory turned out to dissatisfy this principle. But what did Einstein understand by "Mach's principle"? And if you examine Mach's text, you may be puzzled, since there may be several candidates for this name. Anyway, it should be instructive to check the original text by Mach where he states the ideas which seem to be closely related to this "principle".

See also Einstein's comment on Mach, as a philosopher, in 1916.

Ernst Mach

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The significance of personalities like Mach lies by no means only in the fact that they satisfy the philosophical needs of their times, an endeavor which the hard-nosed specialist may dismiss as a luxury. Concepts that have proven useful in ordering things can easily attain an authority over us such that we forget their wordly origin and take them as immutably given. They are then rather rubber-stamped as a "sine-qua-non of thinking" and an "a priori given", etc. Such errors make the road of scientific progress often impassable for long times. Therefore, it is not at all idle play when we are trained to analyze the entrenched concepts, and point out the circumstances that promoted their justification and usefulness and how they evolved from the experience at hand. This breaks their all too powerful authority. They are removed when they cannot properly legitimize themselves; they are corrected when their association with given things was too sloppy; they are replaced by others when a new system can be established that, for various reasons, we prefer. (Einstein, "Ernst Mach", Physikalishe Zeitschrift 17 (1916), 102; Collected Papers vol. 6, Doc. 29)


The core of "Mach's Principle" is something like this: the inertia of a body is determined in relation to all other bodies in the universe (in short, "matter there governs inertia here"). But "inertia" here is ambiguous: inertial mass, or the property expressed by the law of inertia?

Mach states such ideas in the following parts, after he presented his famous objections against Newton's argument for absolute space.

If, in a material spatial system, there are masses with different velocities, which can enter into mutual relations with one another, these masses present to us forces. We can only decide how great these forces are when we know the velocities to which those masses are to be brought. Resting masses too are forces if all the masses do not rest. ... All masses and all velocities, and consequently all forces, are relative. There is no decision about relative and absolute which we can possibly meet, to which we are forced, or from which we can obtain any intellectual or other advantage. (Mach, The Science of Mechanics, ch.2, vi-3, Open Court, 1960, 279)

Here, Mach talks about inertial mass, but his emphasis is on the relativity of motion. In order to understand Mach's assertion, you have to remember that Mach tries to define the notion of mass in terms of acceleration and the principle (law) of reaction. For this, see ch.2, iv and v.

All those bodies are bodies of equal mass, which, mutually acting on each other, produce in each other equal and opposite accelerations. (ch.2, v-3, 266)

Thus, adopting the notion of mass definable along this line, given two bodies A and B, only the ratio of the masses of A and B can be defined for Mach; the concept of mass is relative in this sense.

However, Mach then comes to an argument related to the law of inertia.

The comportment of terrestrial bodies with respect to the earth is reducible to the comportment of the earth with respect to the remote heavenly bodies. If we were to assert that we knew more of moving objects than this their last-mentioned, experimentally-given comportment with respect to the celestial bodies, we should render ourselves culpable of a falsity. When, accordingly, we say, that a body preserves unchanged its direction and velocity in space, our assertion is nothing more or less than an abbreviated reference to the entire universe. (ch.2, vi-6, 285-6)

The considerations just presented show, that it is not necessary to refer the law of inertia to a spacial absolute space. On the contrary, it is perceived that the masses that in the common phraseology exert forces on each other as well as those that exert none, stand with respect to acceleration in quite similar relations. We may, indeed, regard all masses as related to each other. That accelerations play a prominent part in the relations of the masses, must be accepted as a fact of experience; which does not, however, exclude attempts to elucidate this fact by a comparison of it with other facts, involving the discovery of new points of view. (ch.2, vi-8, 288)

Here, Mach denies the need for Newton's absolute space for expressing the law of inertia; and notice that this argument can be extended also against assuming an inertial frame. Mach is arguing, "How can we determine such a frame? Only by referring to other bodies in the universe." This position seems to be basic to Mach.


According to Einstein's formulation (1918), however, Mach's Principle is this:

The G-field is without remainder determined by the masses of bodies. Since mass and energy are, according to results of the special theory of relativity, the same, and since energy is formally described by the symmetric energy tensor (T), this therefore entails that the G-field be conditioned and determined by the energy tensor. (Translation by C. Hoefer, Barbour & Pfister 1995, 67)

What this means is that the metric field of the spacetime is completely determined by the mass-energy distribution in the universe; since the metric field determines the geometry and hence the geodesics (motion along a geodesic is a substitute for an inertial motion). Thus, as a consequence of this, spacetime without any mass-energy distribution becomes meaningless. So far, this seems in accordance with the spirit of Mach's idea (but Minkowski space becomes incompatible with this principle). However, it is quite hard to see the trace of Mach's original motivation, that is, the idea of reconstructing mechanics only in terms of relative motions, without presupposing an absolute space, or an inertial frame for that matter, in the first place. Instead, Einstein often uses an obscure expression "the relativity of inertia", which may, or may not be related to Mach's original idea. Thus, some of the contemporary Machians (such as Julian Barbour) argues that Einstein chose a wrong way to incorporate the Machian idea into the theory of general relativity. But in order to assess such claims, we have got to examine Mach's and Einstein's works in more detail.


The word "Mach's Principle" was, as is already pointed out, introduced by Einstein in 1918, and thanks to Einstein's reputation, many physicists and philosophers began to discuss it; but most of them followed Einstein's rendering(s), rarely going back to Mach's own words.

"Prinzipielles zur allgemeinen Relativitaetstheorie", Annalen der Physik 55, 240-244.

However, there are a number of puzzles as regards Einstein's ambiguity, and as regards Mach's own view in writing down those passages to which Einstein and others refer: what exactly did he mean? These are not an easy question, and interested readers are referred to the Volume 6 of Einstein Studies,

J. Barbour and H. Pfister, eds., Mach's Principle, Birkhaeuser (1995);

a number of eminent scholars discuss this question and related problems. Julian Barbour, in particular, argues that there is a second Mach's Principle with respect to time. This book is valuable also because many hitherto unknwn attempts at a relational theory of classical mechanics, along the line suggested by Mach, are extracted and reproduced. As to the question how far Einstein's general relativity incorporated the Machian ideas, there is no general agreement among the scholars. If you wish to discuss "Mach's Principle", this volume is certainly one of the "compulsory" readings. See also Newsletter 42.

In order to follow some of the more technical materials in this volume, maybe you should consult chapter 21 of Misner et al, Gravitation (1973); section 21.12 in particular has the title "Mach's Principle and the Origin of Inertia". There, the authors give their updated version of Mach's Principle, and sketch a possible line for fulfilling this principle.


Last modified November 18, 2002. (c) Soshichi Uchii
suchii@bun.kyoto-u.ac.jp